ADDENDUM. 



47 



TO NOTE 1, PACK 38. 



Tire apparent antagonism between the theorem of the text and that of Laplaee suggests a f cw 

 additional words. Tin- tlimr.ni uf Laplace is that "in whatever manner the waters of the ocean 

 act iipun tin- earth, either liy their attraction, their pressure, their friction, or by the various resist- 

 which they suffer, they communicate to the axis of the earth a motion which ia very nearly 

 to that it would acquire from the action of tlu sun and moon upon the sea, if it form a solid 

 mass with the earth." (Mcc. Cel., Bowditch [3345.]) 



The theorem is demonstrated in two distinct, quite different, manners. The last demonstration is 

 founded upon the principle of the "conservation of areas;" and as the result of tins demonstration 

 the proposition is stated in the above quoted words. 



The tir-t demonstration is purely analytical, and, after stating that "this fluid" (i. e. of the ocean) 

 " acts upon the terrestrial spheroid by its pressure and by its attraction," Laplace proceeds to find 

 the analytical expressions for the precession and nutation-producing couples due to this pressure and 

 to this attraction as they are modified by the attraction of the sun and moon upon the fluid. Ho 

 then proceeds to calculate these couples for the material substance of the ocean, considered as rigidly 

 connected (or forming a solid mass) with the earth. lie finds the couples, so calculated, respectively, 

 identical in the two cases, and epitomizes the result as follows: "the phenomena of the precession 

 of the equinoxes and the nutation of the earth's axis, are exactly the same as if the sea form a solid 

 mass with the spheroid which it covers." [3287.] 



But this demonstration is limited by the assumption that " the sea wholly covers the terrestrial 

 spheroid or nucleus, that is of a regular depth, and suffers no resistance from the nucleus ;" and both, 

 demonstrations imply an ocean of (relatively) small depth. 



Under the last mentioned treatment of the subject the proposition of Laplace and that which I 

 demonstrate are but the extreme phases exhibited by the solution of a problem, according as the 

 datum be that the depth of the sea is minute (in which case its entire precession-producing couple, 

 not effectively exerted upon its own mass, is almost wholly transferred to the solid nucleus); or 

 that the nucleus is very small, in which case the lost precession-producing couple of the fluid is but 

 in small part transferred to the nucleus or wholly disappears with the vanishing of the latter. 



I think, however, that the last-mentioned (first in point of order) demonstration of Laplace is not 

 as general as the language quoted [3287] would indicate. The omission of variations of the radius- 

 vector, R, in all the integrations gives rise to errors which do not seem to me to be identical in the 

 two processes by which the couples are calculated ; and in the second calculation the errors thus in- 

 volved are commensurable, when the variation of depth is not great, with the quantities sought In 

 expressions [3284] [3285] all manifestation of the ellipticity of the inner surface of the solid shell 

 is lacking ; yet upon this ellipticity, as much as upon the variation of the shell thickness -, , the couples 

 depend, when the latter variation is relatively small. 



An apt illustration of the above remarks is derived from the supposition as admissible as any 

 other that y is constant. In this case, whatever be the ellipiicily of the solid nucleus, the value 

 of y (the height of the diurnal or precession-affecting tide, see [2253] [3333] and also p. 36) is zero, 

 and, of course, the couples [3272] and [3273] become zero as will be found by performing the inte- 

 grations in those equations. So, of course, do expressions [3284] and [3285] become zero, with y 

 made constant. But these list should not be zero except for the case of sphericity of the nucleus. 1 

 The expressions do not seem to me capable of giving with sufficient accuracy the general values of 

 the couples, or of sustaining the inference which I have quoted [3287], even though otherwise 

 tenable 



1 A shell of slight Internal ellipticity ami small uniform thickness has a processional coefficient of half the value 

 of that of a shell bounded by surfaces of equal ellipticity. 



